Wigner surmise type evaluation of the spacing distribution in the bulk of the scaled random matrix ensembles

Random matrix ensembles with orthogonal and unitary symmetry correspond to the cases of real symmetric and Hermitian random matrices respectively. We show that the probability density function for the corresponding spacings between consecutive eigenvalues can be written exactly in the Wigner surmise type form a(s)e −b(s) for a simply related to a Painlevé transcendent and b its anti-derivative. A formula consisting of the sum of two such terms is given for the symplectic case (Hermitian matrices with real quaternion elements). It is well established that universal features of the spectrum of classically chaotic quantum systems are correctly described by random matrix ensembles of an appropriate symmetry [1, 2, 3]. Generically there are three symmetry classes corresponding to two distinct time reversal symmetries plus the situation in which time reversal symmetry is absent. The level repulsion — which is a characteristic of the spectra of chaotic systems and is not present in the spectra of integrable systems — differs for each of the symmetry classes. Thus let p(s) denote the probability density function for the spacing between consecutive levels, calculated after the energy levels are first rescaled so that the mean spacing is unity. Then p(s) ∝ s for a time reversal symmetry T such that T 2 = 1, p(s) ∝ s 2 in the absence of time reversal symmetry while p(s) ∝ s 4 for a time reversal symmetry such that T 2 = −1. It is therefore convenient to distinguish the three cases by the label β and so write p β (s), where β = 1, 2 or 4 depending on the small s behaviour of p β (s). In this work succinct expressions for p β (s) will be given in terms of Painlevé transcendents. The full probability density function p β (s) is by far the most studied statistic in relation to empirical data. For example in the early work [4] (this article is reprinted in [1]) on the energy levels of complex nuclei one sees in Figure 3 an empirical bar graph of p 1 (s) obtained from experimental data plotted on the same graph as a theoretical approximation to p 1 (s) known as the Wigner surmise. This approximation is given by the functional form p W 1 (s) = (πs/2)e −(πs/2) 2. The fact that the Wigner surmise is an approximation rather than exact was soon realized [5], and the task of …